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Intro to factors & divisibility

Learn what it means for polynomials to be factors of other polynomials or to be divisible by them.

What we need to know for this lesson

A monomial is an expression that is the product of constants and nonnegative integer powers of x, like 3x2. A polynomial is an expression that consists of a sum of monomials, like 3x2+6x1.

What we will learn in this lesson

In this lesson, we will explore the relationship between factors and divisibility in polynomials and also learn how to determine if one polynomial is a factor of another.

Factors and divisibility in integers

In general, two integers that multiply to obtain a number are considered factors of that number.
For example, since 14=27, we know that 2 and 7 are factors of 14.
One number is divisible by another number if the result of the division is an integer.
For example, since 153=5 and 155=3, then 15 is divisible by 3 and 5. However, since 94=2.25, then 9 is not divisible by 4.
Notice the mutual relationship between factors and divisibility:
Since 14=27 (which means 2 is a factor of 14), we know that 142=7 (which means 14 is divisible by 2).
14=272 is a factor of 14142=714 is divisible by 2
In the other direction, since 153=5 (which means 15 is divisible by 3), we know that 15=35 (which means 3 is a factor of 15).
153=515 is divisible by 315=353 is a factor of 15
This is true in general: If a is a factor of b, then b is divisible by a, and vice versa.

Factors and divisibility in polynomials

This knowledge can be applied to polynomials as well.
When two or more polynomials are multiplied, we call each of these polynomials factors of the product.
For example, we know that 2x(x+3)=2x2+6x. This means that 2x and x+3 are factors of 2x2+6x.
Also, one polynomial is divisible by another polynomial if the quotient is also a polynomial.
For example, since 6x23x=2x and since 6x22x=3x, then 6x2 is divisible by 3x and 2x. However, since 4x2x2=2x, we know that 4x is not divisible by 2x2.
With polynomials, we can note the same relationship between factors and divisibility as with integers.
2x(x+3)=2x2+6x2x is a factor of 2x2+6x2x2+6x2x=x+32x2+6x is divisible by 2x
6x23x=2x6x2 is divisible by 3x3x(2x)=6x23x is a factor of 6x2
In general, if p=qr for polynomials p, q, and r, then we know the following:
  • q and r are factors of p.
  • p is divisible by q and r.

Check your understanding

1) Complete the sentence about the relationship expressed by 3x(x+2)=3x2+6x.
x+2 is
3x2+6x, and 3x2+6x is

2) A teacher writes the following product on the board:
Miles concludes that 3x2 is a factor of 12x3.
Jude concludes that 12x3 is divisible by 4x.
Who is correct?
Choose 1 answer:

Determining factors and divisibility

Example 1: Is 24x4 divisible by 8x3?

To answer this question, we can find and simplify 24x48x3. If the result is a monomial, then 24x4 is divisible by 8x3. If the result is not a monomial, then 24x4 is not divisible by 8x3.
Since the result is a monomial, we know that 24x4 is divisible by 8x3. (This also implies that 8x3 is a factor of 24x4.)

Example 2: Is 4x6 a factor of 32x3?

If 4x6 is a factor of 32x3, then 32x3 is divisible by 4x6. So let's find and simplify 32x34x6.
Notice that the term 8x3 is not a monomial since it is a quotient, not a product. Therefore we can conclude that 4x6 is not a factor of 32x3.

A summary

In general, to determine whether one polynomial p is divisible by another polynomial q, or equivalently whether q is a factor of p, we can find and examine p(x)q(x).
If the simplified form is a polynomial, then p is divisible by q and q is a factor of p.

Check your understanding

3) Is 30x4 divisible by 2x2?
Choose 1 answer:

4) Is 12x2 a factor of 6x?
Choose 1 answer:

Challenge problems

5*) Which of the following monomials are factors of 15x2y6 ?
Not Factor

6*) The area of a rectangle with height x+1 units and base x+4 units is x2+5x+4 square units.
Which of the following are factors of x2+5x+4?
Choose all answers that apply:

Why are we interested in factoring polynomials?

Just as factoring integers turned out to be very useful for a variety of applications, so is polynomial factorization!
Specifically, polynomial factorization is very useful in solving quadratic equations and simplifying rational expressions.
If you'd like to see this, check out the following articles:

What's next?

The next step in the factoring process involves learning how to factor monomials. You can learn about this in our next article.

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